[過去ﾛｸﾞ] 現代数学の系譜　カントル　超限集合論 (1002ﾚｽ)
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627(1): 現代数学の系譜 雑談 ◆e.a0E5TtKE 2019/12/07(土)14:51 ID:H2e5WMAT(7/14)調 AAS
>>626

つづき

https://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem
Lowenheim?Skolem theorem
(抜粋)
The proof of the upward part of the theorem also shows that a theory with arbitrarily large finite models must have an infinite model; sometimes this is considered to be part of the theorem.

Many consequences of the Lowenheim?Skolem theorem seemed counterintuitive to logicians in the early 20th century, as the distinction between first-order and non-first-order properties was not yet understood.
One such consequence is the existence of uncountable models of true arithmetic, which satisfy every first-order induction axiom but have non-inductive subsets.

Another consequence that was considered particularly troubling is the existence of a countable model of set theory, which nevertheless must satisfy the sentence saying the real numbers are uncountable.
This counterintuitive situation came to be known as Skolem's paradox; it shows that the notion of countability is not absolute.

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628(1): 現代数学の系譜 雑談 ◆e.a0E5TtKE 2019/12/07(土)14:54 ID:H2e5WMAT(8/14)調 AAS
>>626-627

(引用開始)
レーヴェンハイム−スコーレムの定理
定理の上方部分の証明は、いくらでも大きな有限のモデルを持つ理論は無限のモデルを持たねばならないことをも示す。
The proof of the upward part of the theorem also shows that a theory with arbitrarily large finite models must have an infinite model; sometimes this is considered to be part of the theorem.
(引用終り)

後者関数の繰り返し適用で、無限集合ができる
それは、ノイマンの後者関数であれ、ZERMELOの後者関数（＝多重シングルトン）であれ、同じことだよ

無理するな

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